In current practice, there are two types of control algorithms for solid-state over-current relays, namely, electromechanical relay algorithm [1][2] and l2t algorithm. The electromechanical relay algorithm is derived from the model of electromechanical relay using the following:
                                                                                          For                                  ⁢                0                            <              M              <              1                                                                                          t                ⁡                                  (                  I                  )                                            =                                                t                  r                                                                      M                    2                                    -                  1                                                                                A1                                                                              For                ⁢                                                                  ⁢                M                            >              1                                                                                          t                ⁡                                  (                  I                  )                                            =                                                A                                                            M                      P                                        -                    1                                                  +                B                                                                A2      
Where                t(I) is the reset time in Eq. (A1) or trip time in Eq. (A2) in seconds,        M is the Iinput/Ipickup (Ipickup is the relay current set point),        tr is the rest time (for M=0) defined in IEEE Std C37.112-1996 Table 1,        A, B, constants defined in IEEE Std C37.112-1996        p Table 1 to provide selected curve characteristics.        
The relay will trip if
                                          ∫            0                          T              p                                ⁢                                    1                              t                ⁡                                  (                  I                  )                                                      ⁢                          ⅆ              t                                      >        1                    A3      
The discrete form of equation A3 is
                                          ∑                          k              =              0                                      k              =              n                                ⁢                                    Δ              ⁢                                                          ⁢              t                                      t              ⁡                              (                                  I                  k                                )                                                    >        1                    A4      
Where                Tp trip time in seconds,        Δt Sample period in seconds,        t(Ik) t(I) calculated from Eq. (A1) or Eq. (A2) for kth sample of current I.        
For 0<M<1, t(Ik) is a negative number. If the summation in equation A4 keeps going, the sum value will go to—∞. Because this algorithm simulates the reset dynamics of electromechanical relays, the summation shall be stopped if
                                          ∑                          k              =              0                                      k              =              n                                ⁢                                    Δ              ⁢                                                          ⁢              t                                      t              ⁡                              (                                  I                  k                                )                                                    <        0                    A5      
This is equivalent to saying that a motor will reach the same equilibrium temperature without regard to what state the motor is operating. The algorithm does not take into account whether the motor is operating at 50% IFLA or at 80% IFLA. In reality, a motor reaches different equilibrium temperatures when different currents are supplied to the motor. Therefore equation A1 does not simulate motor thermal dynamics for 0<M<1, which results in the algorithm not accurately tracking the used thermal capacity of AC motor under varying load.
The l2t algorithm uses a locked rotor current ILR and safe stall time tLR as motor thermal limit. The cold trip time ttrip-C and the hot trip time ttrip-H for currents above pickup are defined by
                              t                      trip            -            C                          =                                            (                                                I                  LR                                I                            )                        2                    ⁢                      t            LRC                                      B1                                    t                      trip            -            H                          =                                            (                                                I                  LR                                I                            )                        2                    ⁢                      t            LRH                                      B2      
The used thermal capacity θn is calculated by
                              θ          n                =                                            Δ              ⁢                                                          ⁢              t                                      t              trip                                +                      θ                          n              -              1                                                  B3      
There is a θhot or thot to determine the switch of ttrip from ttrip-c to ttrip-H. Note that equation B3 is usually implemented in DSP and is updated only when M>1. There is no update of θn for M<1 so this algorithm does not consider the cooling effect of 0<M<1.
The Amd2 [3] of IEC 60947-4-2 [4] imposes new thermal memory test requirements on solid-state relay protecting AC induction motors, which state: electronic overload relays shall fulfill the following requirements (note table and figures references are provided in Amendment 2 to standard IEC 60947-4-2):                apply a current equal to Ie until the device has reached the thermal equilibrium;        interrupt the current for a duration of 2×Tp (see Table 2 of [3]) with a relative tolerance of ±10% (where Tp is the time measured at the D current according to Table 3 of [3]);        apply a current equal to 7.2×Ie; and        the relay shall trip within 50% of the time Tp.        
From the analysis above, it is seen that both electromechanical relay algorithm and l2t algorithm may fail the thermal memory test and may not provide sufficient protection to AC induction motors. Accordingly, a need exists for a device, method, and system relay algorithm based on motor thermal model. The algorithm may accurately track a motor's used thermal capacity when the motor's current varies at any value satisfying M≧0.